Goals from Moses Odubajo and Scott Cuthbert sealed a ninth away win of the season for Russell Slade's high-flyers on Tuesday night.
Shrewsbury made much of the early running at the New Meadow with Aaron Wildig firing straight at debutant Orient keeper Eldin Jakupovic from distance before spurning a better opportunity when he lifted a shot over the bar after being set free by Tom Eaves.
Orient grew more dangerous as the first half developed and forged a 27th minute lead through promising right-sided player Odubajo.
Striker Jamar Loza, making his debut on loan from Norwich, threaded a clever pass forward and Odubajo coolly rounded keeper Chris Weale before finding the net from a tight angle.
Shrewsbury, who have now lost five successive home games, should have equalised early in the second half but Eaves dragged his shot wide with only the keeper to beat.
Orient always looked dangerous going forward and finally made the game safe three minutes from time when Cuthbert headed in a corner from Dean Cox.